Search results
Results from the WOW.Com Content Network
The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell. The unit-radius 600-cell has tetrahedral cells of edge length 1 φ {\textstyle {\frac {1}{\varphi }}} , 20 of which meet at each vertex to form an icosahedral pyramid (a 4 ...
where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. [2] Any convex polyhedron's surface has Euler characteristic = + = . This equation, stated by Euler in 1758, [3] is known as Euler's polyhedron formula. [4]
Construction from the vertices of a truncated octahedron, showing internal rectangles. The Cartesian coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio [11]
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating triangles and pentagons.
Types of truncation on a square, {4}, showing red original edges, and new truncated edges in cyan. A uniform truncated square is a regular octagon, t{4}={8}. A complete truncated square becomes a new square, with a diagonal orientation. Vertices are sequenced around counterclockwise, 1-4, with truncated pairs of vertices as a and b.
The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a snub octahedron, as s{3,4} or , and seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube.
The vertices of Jessen's icosahedron may be chosen to have as their coordinates the twelve triplets given by the cyclic permutations of the coordinates (,,). [1] With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have length 6 {\displaystyle {\sqrt {6}}} , and the long (reflex) edges have ...